Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree

In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R2 of degree d that in complex notation z=x+iy can be written asz˙=(λ+i)z+Az(d−n+1)/2z¯(d+n−1)/2+Bz(d+n+1)/2z¯(d−n−1)/2+Cz(d+1)/2z¯(d−1)/2+Dz(d−(2+j)n+1)/2z¯(d+(2+j)n−1)/2, where j is either 0 or 1. If j=0 then d⩾5 is an odd integer and n is an even integer satisfying 2⩽n⩽(d+1)/2. If j=1 then d⩾3 is an integer and n is an integer with converse parity with d and satisfying 0